## Eighteenth NWDR-Workshop

**Friday, 22 July 2016**

## Location

At Ruhr-University Bochum (Travel Information).**Talks**: Will be given in lecture hall NA 01/99. **Coffee breaks**: Will take place in room NA 1/58 (Friedrich-Sommer-Raum).

Please find a campus map here.**Dinner**: 19:00 at Amalfi-Pizzeria, Gerberstraße 2, 44787 Bochum.

#### Held in conjuction with

Floer Center for Geometry, Ruhr-University Bochum

Department of Mathematics, Ruhr-University Bochum

## Speakers

Arkady Berenstein (Eugene): Hecke-Hopf algebras

Joseph Bernstein (Tel Aviv): Stacks in Representation Theory --- how should we think about continuous representations of algebraic groups

Grzegorz Bobinski (Torun): Derived classification of the gentle two-cycle algebras

Lennart Galinat (Cologne): Geometric Aspects of the Classical Yang-Baxter Equation

Alexander Kleshchev (Eugene): RoCK blocks of symmetric groups and Hecke algebras

## Schedule

10:30 - 11:00 | Coffee |
---|---|

11:00 - 12:00 | L. Galinat |

12:00 - 13:00 | A. Berenstein |

13:00 - 14:30 | Lunch |

14:30 - 15:30 | J. Bernstein |

15:30 - 16:30 | G. Bobinski |

16:30 - 17:00 | Coffee |

17:00 - 18:00 | A. Kleshchev |

19:00 | Dinner |

## Registration

If you plan to attend, please send an email to the organizers before 30 June (and indicate whether you will attend the joint dinner).

## Abstracts

A. Berenstein: Hecke-Hopf algebras

Abstract: It is well-known that Hecke algebras H_q(W) of Coxeter

groups W do not have interesting Hopf algebra structures because,

first, the only available one would emerge via an extremely complicated

isomorphism with the group algebra of W and, second, this

would make H_q(W) into yet another cocommutative Hopf algebra.

The goal of my talk (based on joint work with D. Kazhdan) is to extend

each Hecke algebra H_q(W) to a non-cocommutative Hopf algebra (we call

it Hecke-Hopf algebra of W) that contains H_q(W) as a coideal

subalgbera.

Hecke-Hopf algebras have a number of remarkable properties: they

generalize Bernstein presentation of Hecke algebras, provide new

solutions to the quantum Yang-Baxter equation and a large class of

endo-functors of the category H_q(W)-Mod, and suggest further

generalizations of Hecke algebras.

J. Bernstein: Stacks in Representation Theory -- how should we think about continuous representations of algebraic groups

G. Bobinski: Derived classification of the gentle two-cycle algebras

Abstract: According to a result of Schröer and Zimmermann the gentle algebras are closed with respect to the derived equivalence. The tree gentle algebras are precisely the algebras derived equivalent to the Dynkin algebras of type A and their derived classification is well known. Similarly, the derived classification of one-cycle gentle algebras is known. In both cases the derived equivalence classes are determined by the invariant introduced by Avella-Alaminos and Geiss. Using this invariant Avella-Alaminos and, independently, Malicki and the speaker obtained partial derived classification of the gentle two-cycle algebras. In the talk we complete this classification.

Lennart Galinat (Cologne): Geometric Aspects of the Classical Yang-Baxter Equation

Abstract: In my talk, which is based on joint work with Igor Burban and Alexander Stolin, I will explain a connection between certain sheaves of Lie algebras on algebraic curves and solutions of the classical Yang-Baxter equation. In particular, I will show how the representation theory of vector bundles on the nodal cubic curve leads to certain explicitly computable quasi-trigonometric solutions in the sense of Khoroshkin-Pop-Samsonov-Stolin-Tolstoy. Furthermore, I will describe a close connection between the rational solutions of Stolin and the cuspidal cubic curve.

Alexander Kleshchev (Eugene): RoCK blocks of symmetric groups and Hecke algebras

Abstract: We present a joint result with Anton Evseev, which describes every block of a symmetric group up to derived equivalence as a certain Turner double algebra. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Brou{\'e} abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. This description was conjectured by Will Turner.

It relies on the work of Chuang-Kessar and Chuang-Rouquier. (RoCK=Rouquier+Chuang+Kessar).

Key idea is a connection with Khovanov-Lauda-Rouquier algebras and their semicuspidal representations.

## Organizers

M. Reineke (markus dot reineke at rub dot de), G. Röhrle (gerhard dot roehrle at rub dot de), 20 July 2016